오일러-가우스 초기하함수2F1

개요

• 초기하급수\[\,_2F_1(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, |z|<1\]

여기서 \((a)_n=a(a+1)(a+2)...(a+n-1)\)에 대해서는 포흐하머 (Pochhammer) 기호 항목 참조

• 적분표현\[\,_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(c-a)\Gamma(a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\,dt\]
• 초기하급수의 해석적확장을 통해 얻어진 함수를 초기하함수라 함
• 오일러, 가우스, 쿰머, 리만,슈워츠 등의 연구

초기하급수로 표현되는 함수의 예

• 많은 special function 은 초기하함수의 파라메터를 변화시켜 얻어짐
• 제1종타원적분 K (complete elliptic integral of the first kind)\[K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)\]
• 제2종타원적분 E (complete elliptic integral of the second kind)\[E(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},-\frac{1}{2};1;k^2)\]

초기하 미분방정식

• \(w(z)=\,_2F_1(a,b;c;z)\) 는 다음 피카드-Fuchs 형태의 미분방정식의 해가 된다

\[z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0\]

• 이 미분방정식을 초기하 미분방정식(Hypergeometric differential equations) 이라 부른다

오일러의 변환 공식

\(_2F_1 (a,b;c;z) = (1-z)^{-a} {}_2F_1 (a, c-b;c ; \frac{z}{z-1})\)

\(_2F_1 (a,b;c;z) = (1-z)^{-b}{}_2F_1(c-a,b;c;\frac{z}{z-1})\)

\(_2F_1 (a,b;c;z) = (1-z)^{c-a-b}{}_2F_1 (c-a, c-b;c ; z)\)

증명

다음 적분표현을 활용

\(\,_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(c-a)\Gamma(a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\,dt\)

위의 우변에서 \(t\to 1-t\), \(t\to \frac{t}{1-z-tz}\), \(t\to \frac{1-t}{1-tz}\)의 변환을 이용하면 항등식이 얻어진다. ■

• http://mathworld.wolfram.com/EulersHypergeometricTransformations.html
• 쿰머의 초기하 미분방정식(Hypergeometric differential equations)에 대한 24개의 해를 표현하는데 사용됨

contiguous 관계

• 초기하함수 2F1의 contiguous 관계

타원적분과 초기하급수

• 제1종타원적분 K (complete elliptic integral of the first kind)\[K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)\]

모듈라 함수와의 관계

• 라마누잔과 파이
• [BB1998]Pi and the AGM
• Jonathan M. Borwein, Peter B. Borwein, Wiley-Interscience (July 13, 1998) 179,180p
• [Nes2002] 159p

슈워츠 s-함수

• 슈워츠 s-함수

special values

• Chu-Vandermonde 공식\[\,_2F_1(-n,b;c;1)=\dfrac{(c-b)_{n}}{(c)_{n}}\] 아래 가우스 공식에서 \(a=-n\)인 경우에 얻어진다

• 가우스 공식\[\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\]
• 위의 두 식에 대해서는 초기하 급수의 합공식
• 렘니스케이트(lemniscate) 곡선과 타원적분\[\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;\frac{1}{2})=K(\frac{1}{\sqrt{2}})=\frac{1}{4}B(1/4,1/4)=\frac{\Gamma(\frac{1}{4})^2}{4\sqrt{\pi}}=1.8540746773\cdots\]
• http://mathworld.wolfram.com/HypergeometricFunction.html\[_2F_1(\frac{1}{3},\frac{2}{3};\frac{5}{6};\frac{27}{32})=\frac{8}{5}\]\[_2F_1(\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{80}{81})=\frac{9}{5}\]\[_2F_1(\frac{1}{8},\frac{3}{8};\frac{1}{2};\frac{2400}{2401})=\frac{2}{3}\sqrt{7}\]\[_2F_1(\frac{1}{6},\frac{1}{3};\frac{1}{2};\frac{25}{27})=\frac{3}{4}\sqrt{3}\]\[_2F_1(\frac{1}{6},\frac{1}{2};\frac{2}{3};\frac{125}{128})=\frac{4}{3}\sqrt[6]2\]\[_2F_1(\frac{1}{12},\frac{5}{12};\frac{1}{2};\frac{1323}{1331})=\frac{3}{4}\sqrt[4]{11}\]\[_2F_1(\frac{1}{12},\frac{5}{12};\frac{1}{2};\frac{121}{125})=\frac{\sqrt[6]{2}\sqrt[4]{15}}{4\sqrt{\pi}}\frac{\Gamma(\frac{1}{3})^3}{\Gamma(\frac{1}{4})^2}(1+\sqrt{3})\]

역사

• 수학사 연표

관련된 항목들

• 주기 (period
• 무리수와 초월수
• 오일러 베타적분
• Special functions
• 맴돌이군과 미분방정식

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxWFFlaHc2OVdQLXc/edit

사전 형태의 자료

• http://ko.wikipedia.org/wiki/초기하함수
• http://en.wikipedia.org/wiki/hypergeometric_functions
• http://en.wikipedia.org/wiki/List_of_hypergeometric_identities
• http://en.wikipedia.org/wiki/hypergeometric_differential_equation
• http://en.wikipedia.org/wiki/Frobenius_solution_to_the_hypergeometric_equation

리뷰논문, 에세이, 강의노트

• [Nes2002]On the Algebraic Independence of Numbers
  • Yu.V. Nesterenko, in A panorama in number theory, or, The view from Baker's garden (by Alan Baker,Gisbert Wüstholz), 2002
• On the Kummer Solutions of the Hypergeometric Equation
  • Reese T. Prosser, The American Mathematical Monthly, Vol. 101, No. 6 (Jun. - Jul., 1994), pp. 535-543
• Ramanujan and hypergeometric and basic hypergeometric series
  • R Askey 1990 Russ. Math. Surv. 45 37-86

관련논문

• http://arxiv.org/abs/1511.00020
• Schrenk, K. J., and J. D. Stevenson. “Numerical Evaluation of the Gauss Hypergeometric Function: Implementation and Application to Schramm-Loewner Evolution.” arXiv:1502.05624 [cond-Mat, Physics:physics], February 19, 2015. http://arxiv.org/abs/1502.05624.
• On the contiguous relations of hypergeometric series
  • Medhat A. Rakha, Adel K. Ibrahim, Journal of Computational and Applied Mathematics, Volume 192, Issue 2, 1 August 2006, Pages 396-410
• Transcendence of periods: the state of the art.
  • M. Waldschmidt., Pure Appl. Math. Q. 2 (2006), 435-463.
• Exceptional sets of hypergeometric series
  • Natália Archinard, Journal of Number Theory Volume 101, Issue 2, August 2003, Pages 244-269
• Thorsley, Michael D., and Marita C. Chidichimo. 2001. “An Asymptotic Expansion for the Hypergeometric Function 2F1(a,b;c;x).” Journal of Mathematical Physics 42 (4) (April 1): 1921–1930. doi:doi:10.1063/1.1353185. http://jmp.aip.org/resource/1/jmapaq/v42/i4/p1921_s1
• Special values of the hypergeometric series III
  • Joyce, G. S.; Zucker, I. J., Mathematical Proceedings of the Cambridge Philosophical Society (2002), 133 : 213-222
• Special values of the hypergeometric series II
  • Joyce, G. S.; Zucker, I. J., Mathematical Proceedings of the Cambridge Philosophical Society (2001), 131 : 309-319
• Special values of the hypergeometric series
  • Joyce, G. S.; Zucker, I. J., Mathematical Proceedings of the Cambridge Philosophical Society (1991) volume: 109 issue: 2 page: 257
• Werte hypergeometrischer funktionen
  • Jürgen Wolfart, Inventiones Mathematicae Volume 92, Number 1 / 1988년 2월

노트

말뭉치

1. This paper introduces the hypergeo package of R routines, for numerical calculation of hypergeometric functions.
2. The package is focussed on ecient and accurate evaluation of the hypergeometric function over the whole of the complex plane within the constraints of xed-precision arithmetic.
3. 2 Numerical evaluation of the Gauss hypergeometric function with the hypergeo package when dened.
4. Writing a, b, c for the two upper and one lower argument respectively, the resulting function 2F1 (a, b; c; z) is known as the hypergeometric function.
5. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al.
6. Elliptic modular functions can sometimes be expressed as the inverse functions of ratios of hypergeometric functions whose arguments a, b, c are 1, 1/2, 1/3, ... or 0.
7. We give a basic introduction to the properties of Gauss hypergeometric functions, with an emphasis on the determination of the monodromy group of the Gaussian hyperegeo- metric equation.
8. Initially this document started as an informal introduction to Gauss hypergeometric functions for those who want to have a quick idea of some main facts on hypergeometric functions.
9. To confuse matters even more, the term "hypergeometric function" is less commonly used to mean closed form, and "hypergeometric series" is sometimes used to mean hypergeometric function.
10. The hypergeometric functions are solutions to the hypergeometric differential equation, which has a regular singular point at the origin.
11. Many functions of mathematical physics can be expressed as special cases of the hypergeometric functions.
12. We consider the asymptotic behavior of the Gauss hypergeometric function when several of the parameters a,b,c are large.
13. Computes the Gauss hypergeometric function 2F1(a,b;c;z) and its derivative for real z, z<1 by integrating the defining differential equation using the Matlab differential equation solver ode15i.
14. The major development of the theory of hypergeometric function was carried out by Gauss and published in his famous work of 1812.
15. Almost all of the elementary functions of Mathematics are either hypergeometric, ratios of hypergeometric functions or limiting cases of a hypergeometric series.
16. Two hypergeometric functions with the same argument z are contiguous if their parameters a, b and c differ by integers.
17. A contiguous relation between any three contiguous hypergeometric functions can be found by combining linearly a sequence of Gauss contiguous relations.
18. In this course we will study multivariate hypergeometric functions in the sense of Gelfand, Kapranov, and Zelevinsky (GKZ systems).
19. These functions generalize the classical hypergeometric functions of Gauss, Horn, Appell, and Lauricella.
20. We end with a brief discussion of the classication problem for rational hypergeometric functions.
21. For one-variable hypergeometric functions this interplay has been well understood for several decades.
22. Abstract The two most commonly used hypergeometric functions are the conflu- ent hypergeometric function and the Gauss hypergeometric function.
23. Except for specific situations, computing hypergeometric functions is difficult in practice.
24. (a)j (b)j zj j ! , which is also commonly denoted by M(a; b; z) and is itself often called the confluent hypergeometric function.
25. The function 2F1(a, b; c; z) is commonly denoted by F (a, b; c; z) and is also fre- quently called the Gauss hypergeometric function.
26. We nd that, for both the conuent and Gauss hypergeometric functions, there is no simple answer to the problem of their computation, and dierent methods are optimal for dierent parameter regimes.
27. 3.3 Writing the conuent hypergeometric function as a single fraction . . . . . .
28. 4.3 Writing the Gauss hypergeometric function as a single fraction .
29. The computation of the hypergeometric function pFq, a special function encountered in a variety of applications, is frequently sought.
30. Returns hyp2f1 scalar or ndarray The values of the gaussian hypergeometric function.
31. Using (41) and subordination property, we have thatBy adopting (13) and (64), we obtainApplying hypergeometric function (59), we obtain the upper bound for the case .
32. Recently, there has been a clear interest on Bessel and hypergeometric functions from the point of view of geometric function theory.
33. A hypergeometric function is the sum of a hypergeometric series, which is dened as follows.
34. When one of the numerator parameters ai equals N , where N is a nonnegative integer, the hypergeometric function is a polynomial in z (see below).
35. Sometimes the most general hypergeometric function pFq is called a generalized hypergeo- metric function.
36. For the hypergeometric function 2F1 we have an integral representation due to Euler: Theorem 1.
37. This generally speeds up evaluation by producing a hypergeometric function of lower order.
38. Euler introduced the power series expansion of the form: where a, b, c are rational functions and F(a, b, c, z) is called the hypergeometric function.
39. The hypergeometric function takes a prominent position amongst the world of standard mathematical functions used in both pure and applied mathematics.
40. Gauss was aware of the multi-valuedness of the hypergeometric functions, known in recent times as the monodromy problem.
41. The modern notation for the Gauss hypergeometric function is according to Barnes (1908).
42. The classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function.
43. This paper claims that the natural logarithm can be represented by the Gaussian hypergeometric function.
44. We establish two quadratic transformations for Gauss hypergeometric function in terms of nite summation of combination of two Clausen hypergeometric functions.
45. Further, we have generalized our quadratic transformations in terms of general double series identities as well as in terms of reduction formulas for Kamp de Friets double hypergeometric function.
46. The Hypergeometric functions of two variables was introduced by Appell (1880) and Lauricella (1893) generalized them to several variables.
47. It is interesting to mention here that the results are very important for the application point of view, whenever hypergeometric functions reduce to gamma functions.
48. In this sequel, using the same technique, we establish certain integral transforms and fractional integral formulas for the generalized Gauss hypergeometric functions .
49. For , and , we have where the , a special case of the generalized hypergeometric function (10), is the Gauss hypergeometric function.
50. Further, if we set and in Theorems 1 to 5 or make use of the result (8), we obtain various integral transforms and fractional integral formulas for the Gauss hypergeometric function .
51. This a hypergeometric equation with constants a, b and c dened by F = c, G = (a + b + 1) and H = ab and can therefore be solved near t = 0 and t = 1 in terms of the hypergeometric function.
52. Problems: Find the general solution of each of the following dierential equations near the indicated singular point in terms of hypergeometric function.
53. However, the hypergeometric function is defined over the whole of the complex plane, so analytic continuation may be used if appropriate cut lines are used.
54. Gausss hypergeometric function gives a modular parame- terization of period integrals of elliptic curves in Legendre normal form E() : y2 = x(x 1)(x ).
55. Legendre elliptic curves, hypergeometric functions.
56. Hypergeometric functions are rarely in a form in which these formulae can be applied directly.
57. Writing sums as hypergeometric functions has the great advantage of simplifying manipulation by computer algebraic methods.
58. It is assumed that the hypergeometric functions are convergent and do not contain negative integers in the bottom parameter list.
59. A particular solution of Gausss hypergeometric differential equation (1) is known as Gausss hypergeometric function or simply hypergeometric function.
60. 2 ​ F 1 ​ ( a , b , c , z ) Regularized Gauss hypergeometric function RisingFactorial ( z ) k \left(z\right)_{k} ( z ) k ​ Rising factorial Pow a b {a}^{b} a b Power Factorial n ! n !
61. The hypergeometric functions are solutions to the Hypergeometric Differential Equation, which has a Regular Singular Point at the Origin.

메타데이터

위키데이터

• ID : Q21028472

Spacy 패턴 목록

• [{'LOWER': 'hypergeometric'}, {'LEMMA': 'function'}]
• [{'LOWER': 'gaussian'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'function'}]
• [{'LOWER': 'ordinary'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'function'}]